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The angular diameter distance is defined by $d_A=x/\theta$ for a small transverse physical size $x$ subtending angle $\theta$. In a spatially flat Friedmann-Lemaître-Robertson-Walker metric,where $\chi$ is the comoving radial distance.

The cosmic microwave background consists of relic photons from the early hot, nearly thermal photon-baryon fluid. Before cosmological recombination, abundant free electrons repeatedly scatter photons by Thomson scattering. Their mean free path ${\lambda }_{\gamma }=(n_e{\sigma }_T{)}^{-1}$ is short and their Thomson scattering rate $n_e{\sigma }_{T}c$ is much greater than the Hubble parameter. An earlier photon therefore cannot stream directly to us: subsequent scattering repeatedly changes its direction.

As cosmic expansion cools the plasma to about $T\simeq 3000\mathrm{K}$, electrons and protons combine into neutral hydrogen. The free-electron abundance drops rapidly, the scattering optical depth to the observer becomes of order one, and photon decoupling occurs. Since the cosmic microwave background temperature scales as $T=T_0(1+z)$, with $T_0\simeq 2.725\mathrm{K}$, this occurs at $z\simeq 1100$. The small baryon-to-photon ratio delays neutralization to well below $13.6\mathrm{eV}/k_B$: the high-energy tail contains enough photons to ionize hydrogen until the temperature is much lower than the Hydrogen binding energy expressed as a temperature. The observed cosmic microwave background is primarily this pre-existing radiation undergoing its last scattering; its nearly blackbody spectrum is then preserved by cosmological redshift.

For the requested estimate take a flat matter-dominated universe, $a(t)=(t/t_0{)}^{2/3}$. Restore $c$ in distances. The comoving particle horizon at cosmological recombination and the comoving radial distance to the Cosmic microwave background last-scattering surface areThe proper particle horizon is $a_{\ast }{r}_{\mathrm{hor},\ast }=3c{t}_{\ast }$ and the angular diameter distance is $D_A=a_{\ast }{\chi }_{\ast }$. Therefore the angular particle-horizon size in a power-law universe, taking one horizon radius as the transverse length, isThe angular diameter of a patch of that radius is about $3.56^{\circ }$. The factor of two is a radius-versus-diameter convention; either gives a horizon scale of only a few degrees. This is a particle horizon, rather than the smaller sound horizon. Including early radiation in cosmology changes the numerical estimate but preserves its order of magnitude.

The horizon problem is that the cosmic microwave background has almost the same temperature across the whole sky, including pairs of regions whose past light cones do not overlap in this noninflating model. Local thermal equilibration cannot explain that observation within this history. A sufficiently long period of cosmic inflation makes the comoving Hubble radius decrease: a region once small enough to share a causal past can expand to encompass our entire observed Cosmic microwave background last-scattering surface. After reheating, it can be homogeneous across many later particle horizon patches. The matter-only estimate integrates from a decelerating Big Bang; extending the earlier history by cosmic inflation changes that causal conclusion.

In an Einstein-de Sitter universe, take $a(t)=(t/t_0{)}^{2/3}$ and $t_0=2/(3H_0)$. The galaxy seen today at $z=1$ emitted the light now reaching us when $a(t_e)=1/2$, so $t_e/t_0=2^{-3/2}$. Its fixed comoving radial distance isPut $b=1-1/\sqrt{2}$. A new message sent now reaches the galaxy at $t_1$, whereAssume an immediate reply and negligible local handling delays. The return consumes the same additional conformal time, giving the Einstein-de Sitter round-trip message timeFor the numerical value $H_0=72\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ supplied later in the paper, $H_0^{-1}=13.58\mathrm{Gyr}$ and the waiting time is about $27.1\mathrm{Gyr}$. The age $t_0=9.05\mathrm{Gyr}$ here belongs to the stipulated matter-only model, not a claim about the actual measured cosmic age.

After $n$ one-way legs, $t_n=t_0(1+nb{)}^3$. Since ${\int }_{t_0}^{\infty }dt/a(t)=\infty$, every finite number of messages is possible and there is no finite fundamental upper count. The exchange takes arbitrarily long, with successively longer intervals; infinitely many messages are not completed in finite cosmic time.

Take the expanding flat branch $a(t)=A{e}^{H_{\Lambda }t}$, with $A>0$. A radial null geodesic in FLRW spacetime satisfies $d\rho /dt=\pm 1/a(t)$. For the incoming past light cone of the spacetime event $(t_0,\rho =0)$, integration giveswith arbitrary angular direction. Equivalently, conformal time $\eta =-1/(H_{\Lambda }a)$ gives $\rho ={\eta }_0-\eta$.

Let the reception time tend to infinity. The largest comoving radial distance from which a signal emitted at time $t$ can reach the central comoving observer isThus the cosmological event horizon has constant proper distance in cosmology:An emitted spacetime event is seen at a finite time if and only if $a(t)\rho <H_{\Lambda }^{-1}$. Equality would require $t_0=\infty$, while an event with $a(t)\rho >H_{\Lambda }^{-1}$ never reaches the observer. This is a bound on events at their emission time; a fixed comoving observer can have earlier visible events and later invisible ones.

The boundary really is a null hypersurface. For $f(t,\rho )=\rho -[H_{\Lambda }a(t){]}^{-1}$, its normal has components ${\partial }_{t}f=1/a$ and ${\partial }_{\rho }f=1$, soThe finite future integral, and hence this observer event horizon, uses the expanding branch. For the contracting flat solution $a=A{e}^{-H_{\Lambda }t}$, the same integral diverges and there is no future observer event horizon of this kind; the static zero-density solution likewise has none.

Use the flat matter-only Einstein-de Sitter approximation for the distance-redshift relation, neglecting radiation corrections at cosmological recombination. Let $z_{\ast }$ be the cosmological recombination redshift. Adiabatic photon cooling gives $1+z_{\ast }=T_{\ast }/T_0$. The Hubble rate is $H(z)=H_0(1+z{)}^{3/2}$, so the comoving radial distance to last scattering isThe angular diameter distance is $D_A={\chi }_{\ast }/(1+z_{\ast })$. A proper Hubble radius $H_{\ast }^{-1}=H_0^{-1}(1+z_{\ast }{)}^{-3/2}$ therefore subtends the small angleThis is the angular Hubble-radius scale in an Einstein-de Sitter universe; the present Hubble constant cancels.

Taking $T_{\ast }=0.3\mathrm{eV}$ and the literal printed $T_0=2.35\mathrm{meV}$ gives $1+z_{\ast }=127.7$ and ${\theta }_H\simeq 0.0486$ radians, or $2.8$ degrees. As noted above, that present CMB energy is too large by ten. With the corrected $T_0=0.235\mathrm{meV}$, $1+z_{\ast }=1277$ andwhich gives the familiar order-one-degree scale within the assumed cosmology. Using $0.28\mathrm{eV}$ to mark a few-per-cent ionization fraction changes this only modestly. These numbers refer to the Hubble-radius scale, not its diameter. In a matter-only universe the particle-horizon radius is $2H^{-1}$ and would give twice the angle; the acoustic sound horizon is a different length again.

Follow successive photons from the same comoving source. Their comoving radial distance is fixed, soIt follows that $d{t}_e/d{t}_0=a(t_e)/a(t_0)=1/(1+z)$. Taking the logarithmic derivative of $1+z=a(t_0)/a(t_e)$, with both endpoints allowed to vary, givesTherefore the observer-time redshift drift isHolding the emission time fixed would omit the second term and would not compare successive observations of the same source.

A radial light ray in the FLRW metric travels comoving radial distance $d\chi =dt/a$ in units $c=1$. The total comoving visibility radius is therefore the full available conformal time interval,Insert the Friedmann equation from part (a), then put $y=1/a$ to getFor $0<{\Omega }_M<1$ the integrand is bounded at zero and decays as $y^{-3/2}$ at infinity, so the total conformal lifetime of a flat matter-Lambda universe is finite. Equivalently, this distance is the present comoving particle horizon plus the remaining comoving distance to the cosmological event horizon. This describes causal visibility under the assumed ability to see back to the initial singularity.

The substitution $u=y[{\Omega }_M/(1-{\Omega }_M){]}^{1/3}$ also evaluates the integral using the beta function:

For a radial null geodesic in a spatially flat universe, the comoving radial distance increment is $d\chi =cdz/H(z)$. If the cloud and quasar redshifts differ only slightly, take $H$ at their common approximate epoch $z_{\ast }$, givingMultiplying by the scale factor at that epoch gives the local proper distance in cosmology relevant to the quasar's radiation flux:If distance is instead expressed in comoving units, it is $(1+z_{\ast })d_{cq}$. Equivalently the small rest-frame velocity difference is $\delta v=c\delta z/(1+z_{\ast })$, and $d_{cq}\simeq \delta v/H(z_{\ast })$. This quasar-cloud redshift separation assumes the redshift difference is due to homogeneous expansion, with negligible peculiar velocities. Since the two observations lie on a light cone, the proper separation is understood as the short separation on a nearly common cosmic-time slice; specifying the epoch matters at higher order.

For a radial null geodesic in FLRW spacetime, the FRW metric gives $cdt/a(t)=\pm dr/\sqrt{1-k{r}^2}$. The emitter and receiver have fixed comoving coordinates, so two successive wave crests satisfy the same comoving radial distance relation:Subtracting the two expressions in the short-wave-period limit yields $\delta t_e/a(t_e)=\delta t_0/a(t_0)$. Comoving observers measure these intervals as proper time; since $\nu =1/\delta t$,For a fixed reception epoch, the cosmological redshift factor is inversely proportional to the emission scale factor. This also shows the associated cosmological time dilation.

A radial null geodesic in the FLRW metric obeys $d\chi =c|dt|/a$. The redshift-time relation $dt=-dz/[(1+z)H(z)]$, with $a_0=1$, therefore givesThis is comoving radial distance. The proper distance in cosmology at the clouds' approximately common epoch is insteadThe distinction is essential when quoting a cloud separation. The approximation neglects the change of $H$ across the interval and assumes the observed spacing is dominated by cosmological redshift; peculiar velocity can perturb the inferred separation. It is the redshift version of proper absorber separation from a wavelength interval.

Expand the fractional CMB temperature fluctuation as $\Theta (\mathbf{n})={\sum }_{\ell m}{a}_{\ell m}{Y}_{\ell m}(\mathbf{n})$. Under statistical isotropy, its CMB angular power spectrum is defined byThe normalization is for dimensionless temperature, rather than temperature in kelvin. The spherical harmonics separate angular scales, with large $\ell$ corresponding to small angles.

For approximately constant $R,\varphi ,\psi$, a useful solution of the photon-baryon acoustic oscillator isHere $r_s$ is the comoving sound horizon. With slowly varying $R$, the oscillatory phase is still $k{r}_s$ and its homogeneous amplitude varies approximately as $(1+R{)}^{-1/4}$ under the short-period approximation. Evolving potentials add gravitational driving rather than erase this phase coherence.

The regular growing adiabatic initial conditions produce an initial density displacement and negligible initial acoustic velocity outside the horizon. They therefore select the cosine phase, with $B$ approximately zero, for all modes. At last scattering, extrema of compression and rarefaction occur at $k{r}_s({\eta }_{\ast })\simeq n\pi$. The observed monopole-plus-potential Sachs-Wolfe combination is ${\Theta }_0+\psi ={\delta }_{\gamma }/4+\psi$, so baryon loading shifts its equilibrium and makes alternating peaks unequal. The velocity produces a phase-shifted Doppler CMB anisotropy.

Projection from the Cosmic microwave background last-scattering surface places a mode near $\ell \simeq k{\chi }_{\ast }$, where ${\chi }_{\ast }={\eta }_0-{\eta }_{\ast }$ is its comoving radial distance in a flat universe. The resulting adiabatic acoustic-peak spacing isCoherent acoustic phases turn successive compressions and rarefactions into the CMB peak series. Finite-width last scattering, Doppler projection, evolving potentials and diffusion shift or broaden actual peaks, so this is the requested approximate spacing, not an exact prediction for every peak.

The Cosmic neutrino background began free streaming at neutrino decoupling, long before cosmological recombination, so its directional flux can retain information about density fluctuations from epochs inaccessible to the cosmic microwave background. Earlier decoupling does not by itself guarantee a larger comoving radial distance. A sufficiently massive neutrino eventually becomes nonrelativistic, and its travelled distance iswhich may be smaller than the photon distance because $v(a)<c$. Hence the cosmic neutrino background last-scattering surface need not lie beyond the Cosmic microwave background last-scattering surface; it generally does for neutrinos that remain sufficiently relativistic for sufficiently long.

For a component with the barotropic equation of state $P=w\rho$, the cosmological perfect-fluid continuity equation isUsing $H=\dot{a}/a$ makes this a separable differential equation:Taking $a_0=1$ gives the constant-equation-of-state density scalingThe spatially flat Friedmann equation and the cosmological density parameters therefore giveAlong a radial light ray, $d\chi =dt/a$. Since the redshift-time relation is $dt=-dz/[(1+z)H]$ and $a=(1+z{)}^{-1}$,Hence the comoving radial distance is

For nearby redshifts in a homogeneous expanding universe, the comoving radial distance increment is $c\delta z/H(z)$. Multiplying by the scale factor at the absorber's epoch gives the displayed proper distance in cosmology. The conversion neglects peculiar velocities; observed redshift differences can also contain such velocities and errors in the quasar emission redshift.

In a spatially flat FLRW metric, the supremum of the comoving radial distance from which a signal emitted at the initial time can eventually reach a fixed comoving observer is the total conformal time interval. At an intermediate time $t$, this equals its comoving particle horizon plus the comoving radius of its cosmological event horizon, when both integrals converge. This is a visibility bound for emission arbitrarily close to the initial boundary; it does not assert that a transparent universe or a source existed there.

The equation of state $p=-\varrho$ makes the cosmological continuity equation imply constant energy density ${\varrho }_0$. For ${\varrho }_0>0$, put $H_{\Lambda }^2=8\pi G{\varrho }_0/3$. Solving both the Friedmann equation and Friedmann acceleration equation givesUp to shifts of cosmic time and time reversal, the positive scale factors are $A{e}^{H_{\Lambda }t}$ for $k=0$, $H_{\Lambda }^{-1}\cosh (H_{\Lambda }t)$ for $k=1$, and $H_{\Lambda }^{-1}\sinh (H_{\Lambda }t)$ on $t>0$ for $k=-1$. They are the flat, closed and open slicings of de Sitter spacetime. The open zero is a coordinate boundary with finite Ricci scalar $12H_{\Lambda }^2$ and finite Kretschmann scalar $24H_{\Lambda }^4$.

For the expanding flat branch, the cosmological event horizon has comoving radial distance ${\int }_t^{\infty }ds/a(s)=1/[H_{\Lambda }a(t)]$ and proper distance in cosmology $H_{\Lambda }^{-1}$. An event on the horizon reaches the central comoving observer only at infinite reception time; events beyond it never do.

Stipulate $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$, so $c/H_0=4282.7494\mathrm{Mpc}$. A megaparsec is $10^6$ parsecs. Neglect radiation in cosmology. Compare an accelerating flat model $({\Omega }_{m,0},{\Omega }_{\Lambda ,0})=(0.3,0.7)$, an open matter-only model $(0.3,0)$ and the Einstein-de Sitter universe $(1,0)$. Their present deceleration parameters are respectively $-0.55$, $0.15$ and $0.5$. For these models,Here $\chi$ is dimensionless; the radial null geodesic in FLRW spacetime supplies its integral and spatial curvature of an FLRW universe supplies $S_k$. The observer-to-source comoving radial distance is $(c/H_0)\chi$, whereas the transverse distance uses $S_k$. Omitting the latter distinction would bias the open-model comparison.

At $z=0.5$, numerical integration gives $\chi =0.440984335$ for the flat accelerating model and $\chi =0.392227378$ for the open matter-only model. ThusThe accelerating model is $0.21559$ mag fainter than the open matter-only prediction at the same redshift and standardized absolute magnitude, corresponding to $1.10438$ times the luminosity distance and $0.81991$ times the radiative flux. These are model calculations, not observations from either collaboration.

The Milne model has $q=0$, ${\Omega }_{m,0}={\Omega }_{\Lambda ,0}=0$ and ${\Omega }_{k,0}=1$, giving $\chi =\log (1+z)$ and $d_L=(c/H_0)(z+z^2/2)$. The Einstein-de Sitter luminosity-distance relation independently checks the flat matter-only integral. At $z=0.5$, the accelerating model exceeds the Milne model by only $0.12317$ mag, illustrating the dependence of a residual on the chosen reference. Over a longer redshift interval, luminosity distance integrates past deceleration as well as recent acceleration; a residual need not increase forever.

Multiplying $H_0$ by $f>0$ at fixed density fractions divides every $d_L$ by $f$ and shifts every distance modulus by $-5{\log }_{10}f$. The predicted apparent magnitude is unchanged if $M_{B,0}$ simultaneously increases by $5{\log }_{10}f$. This is the absolute magnitude–Hubble constant degeneracy. For example, $70\to 77\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ shifts $\mu$ by $-0.20696$ mag and $M_{B,0}$ by $+0.20696$ mag. Relative curves at the same $H_0$ retain their separation; determining $H_0$ requires an external absolute magnitude calibration.